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ADVANCED VIBRATIONS ANDADVANCED VIBRATIONS AND NOISE ENGINEERING

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Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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Introduction to Vibrations

Vibrations are present everywhere in life ------ Atomic vibrations (temperature)

Human body (heart)

Machines (large and small)

E th k tEarth quake tremors

Musical instruments …….

So we live in a world of vibrationsSo we live in a world of vibrations….

Let us , from now onwards look at the world from this view point

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Practical Examples of Vibrations

Everyday examples where vibrations are involved :

Cars BikesCars Bikes

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Practical Examples of Vibrations

5Heavy machinery ( turbines and airplane engines )

Galileo Galilei Italian mathematician his great work in dynamics

Practical Examples of Vibrations

Galileo Galilei ----- Italian mathematician ---- his great work in dynamics

SIMPLE PENDULUMSIMPLE PENDULUM

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Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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Fundamentals of Vibrations

Vibration Analysis:

PHYSICALDYNAMICSYSTEM

MATHEMATICALMODEL

GOVERNINGEQUATIONS

SOLVE GOVERNINGEQUATIONS

INTERPRETERESULTS

Example:

8FBD

Fundamentals of Vibrations

Single degree of freedom (DOF)

Two DOFModeling of the systems

Two DOF

Multi DOF

Continuous system

Each system can be under

Free Damped Forced

9Or a combination of these modes

DOF:

Fundamentals of Vibrations

DOF:In vibrations DOF refers to the number of independent co-ordinates with which we can completely define the system at any point of timewhich we can completely define the system at any point of time

Select DOF for any given system considering:Select DOF for any given system considering:

•Simplicity of analysis

•Validation with real life approximations•Validation with real life approximations

Let us see how this works ----

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Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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SDOF reduction of bike rider system

Mathematical System Modeling

SDOF reduction of bike rider system

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Two DOF modeling of bike rider system

Mathematical System ModelingTwo DOF modeling of bike rider system

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Three (multi) DOF modeling of bike rider system

Mathematical System ModelingThree (multi) DOF modeling of bike rider system

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M d li f bik id

Mathematical System Modeling

More accurate modeling of bike rider system

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Other Examples of vibration systems

Single cylinder Internal Combustion Engine

16Equivalent model representation ( Torsional )

Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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Analysis of vibrating systems

F ib iFree vibrations

When a body is set into vibrations and left to itself , it vibrates about a mean positionposition.

This is called free vibration . And the frequency of vibration is called Natural q yFrequency

Natural Frequency

18Best example

Analysis of vibrating systems

Forced vibrationForced vibration

A body acted upon by an external force of particular frequency responds at that frequency and hence the body under consideration is said to be inat that frequency and hence the body under consideration is said to be in forced vibration

FF

Forced vibrations are very common in mechanical systemssyste s

Eg:

C

Single degree of freedom system with forced vibration

Car

Turbine , turbine blades

Machine shafts19

vibrationMachine shafts

Even earthquake excitation

Analysis of vibrating systems

Vibrations are both useful as well as harmful…… just like coefficient of friction

We prefer to travel in a train than by an old bus…….. (vibrations)

Machines are not supposed to vibrate at more than particular level to prod ce eno gh acc racproduce enough accuracy

Life of the systems depends on the DYNAMICS of the machineLife of the systems depends on the DYNAMICS of the machine

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Analysis of vibrating systems

Simple harmonic motion (SHM)Simple harmonic motion (SHM)

Best example : Simple pendulum

Displacement equation : x Xcos(pt) p frequencyDisplacement equation : x = Xcos(pt) p – frequency

X – amplitude

The variation of displacement velocity and acceleration of a SHM system with time are given in the figure

2

cos( )sin( )

x Xp ptx Xp pt

′ = −

′′ = − sin( )x Xp pt=

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Analysis of vibrating systems

Combining two SHM ‘sCombining two SHM s

When we combine two SHM of almost same frequency are combined , a phenomenon called beats is observed p

1 1 2 2cos sin( )x X p t X p t ϕ= + +

P1=(2 πm)/T P2=(2 πn)/T 1p mp n

=2p n

The resultant signal oscillates with two frequencies

2π 2πand

222p

1pand

Analysis of vibrating systems

Beat Phenomenon

( )1 2cos cosx X p t p t= + p1 and p2 are nearly equal

1 2 1 22 cos cos2 2

p p p px X t t− +=

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Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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Fourier Analysis

Most Practical Excitations are Periodic in NatureMost Practical Excitations are Periodic in Nature

• Unbalance excitation in reciprocating mechanism )2cos(cos2 trtmeFP ωωω +=engine

)(uP

• Gas torque in IC engine

• Displacement excitation in Cam- Follower system

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Fourier Analysis

Periodic Seismic Excitation

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Fourier Analysis

2

1.2

1

2

plitu

de =+

mpl

itude

0.6

0.8

1.0

0 40-1

0Am A

m

0.2

0.4

0.05

0.1

0.15

0.20

10

20

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Frequency

TimeFrequency (Hz)

0 10 20 30 40 50 600.0

Decomposition of time domain periodic signal in frequency domain

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Fourier Analysis

6

8

T

2

4

litud

e

0

2

Am

pl

0 0 05 0 1 0 15 0 2 0 25 0 3 0 35-4

-2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Time(sec)

1. To find different frequency components

282. Amplitudes of different components

0.5

1

1.5

e

RESPONSE ESTIMATION

1 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-1

-0.5

0

Time (sec)

Am

plitu

de

Excitation function to systemf(t)=cos(2πft)+0.33 cos(2π(2f)t)

f=20Hzω=2πf

f2=2f=40Hz0 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Am

plitu

de

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Am

plitu

de

f1=f=20Hzω1=2πf1

Harmonic components of excitation f ti f2 2f 40Hz

ω2=2πf20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.8

Time (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.8

Time (sec)

SYSTEM SYSTEM

function

-0.5

0

0.5

1

1.5

Am

plitu

de

-0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de

Response of the system to these harmonic

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1.5

-1

Time (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.8

-0.6

-0.4

Time (sec)+

components of excitation function

2

2 .5

C l t

-0 .5

0

0 .5

1

1 .5

Am

plitu

de

Complete response of the system

290 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 9 0 .1

-1 .5

-1

T im e (s e c )

The break up of excitation frequency and response of the system to the excitation

Fourier Analysis

Frequency analysis to help detect possible resonance condition

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Fourier Analysis

F i S i

Any periodic function can be expressed as a summation of

Fourier Series

)sin()cos()( tbttf ∑∑∞∞

+

y p pinfinite sum of a series of pure sinusoids

)sin()cos()(10

tnbtnatfn

nn

n ωω ∑∑==

+=

where, the coefficients of the sinusoids are given by

ωπ /2

/2

∫=ωπ

πω /2

00 )(

2dttfa

31∫=ωπ

ωπω /2

0

)cos()( dttntfan and ∫=ωπ

ωπω /2

0

)sin()( dttntfbn

Fourier Analysis

)2i ()i ()( AAA

Amplitude variation of Fourier coefficients with number of terms (n)

.....)2sin(2

)sin(2

)( −−−= tAtAAtx ωπ

ωπ

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Vibration Engineeringb at o g ee g

T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application

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Applications

Modeling of such systems should be done as MDOF34

Modeling of such systems should be done as MDOF

Applications

Big and heavy penstock

B tt b d l dBetter be modeled a MDOF or a continuous system

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Development of Tuned Mass Damper

Applications

Development of Tuned Mass Damper

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Applications

Another important application of vibration analysis is signature analysiswhere

Di ti d h lth it iDiagnostics and health monitoring

Life estimation

Life extensionLife extension

Initial the structures are made robust and vibration data is recorded . After some time or after some effects( like earthquake on bridges) the new vibration data tells us about the changes in the structure

The changes are due to

cracks developed

t t t37

support movement etc

Applications

Health equipment (vibrator38

Health equipment (vibrator for massaging)

Forging HammerApplications

Forging HammerVehicle Dynamic Simulatory

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Assignment1

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Assignment

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Assignment

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